Question

Find the greatest number of four digits which is exactly divisible by 15,24 and 36.

Answer 1

Hint: Here, we will find the L.C.M. of the given three numbers. We will divide the greatest possible four digit number by their LCM. Then by subtracting the remainder from the greatest possible four digit number we get the required greatest number of four digits which is exactly divisible by the given numbers.

Complete step-by-step answer:
First of all, by using the prime factorization method, we will find the factors of the given three numbers.
Hence, prime factorization of the first number 15 is:
$15 = 3 \times 5$
Now, prime factorization of the second number 24 is:
$24 = 2 \times 2 \times 2 \times 3 = {2^3} \times 3$
And, prime factorization of the third number 36 is:
$36 = 2 \times 2 \times 3 \times 3 = {2^2} \times {3^2}$
Now, we will find the LCM of these three numbers.
Hence, we will take all the factors present in three numbers and the highest power of the common factors respectively.
Hence, L.C.M. of these three numbers $ = {2^3} \times {3^2} \times 5 = 8 \times 9 \times 5 = 360$
Therefore, the L.C.M. of 15,24 and 36 is 360
Now, we know that the greatest number of four digits is 9999.
Now, we will divide this number by the LCM of the given three numbers.
Hence, we will use the division algorithm to find the remainder.
Dividend $=$ (Quotient $\times$ Divisor) $+$ Remainder
Substituting the values, we get
9999 $=$ (27 $\times$ 360) $+$ 279
Now, clearly, when 9999 is divided by 360, we are left with the remainder 279.
Hence, we will subtract this remainder from 9999
Thus, we get, $9999 - 279 = 9720$

Therefore, the greatest number of four digits which is exactly divisible by 15,24 and 36 is 9720.
Hence, this is the required answer.

Note:
In this question, we are required to express the given numbers as a product of their prime factors in order to find their LCM. Hence, we should know that prime factors are those factors which are greater than 1 and have only two factors, i.e. factor 1 and the prime number itself.
Now, in order to express the given numbers as a product of their prime factors, we are required to do the prime factorization of the given numbers. Now, factorization is a method of writing an original number as the product of its various factors. Hence, prime factorization is a method in which we write the original number as the product of various prime numbers.